Periodic waves spread out over the surface of a lake where two women, Jenna and Mel, are fishing in separate boats 97 meters apart. Each woman's boat bobs up and down 31/min. At a time when Jenna's boat is at a crest, Mel's boat is at its lowest point, and there are 4 additional crests between them. Calculate the wavelength of these water waves.
To calculate the wavelength of the water waves, we need to consider the distance between two consecutive crests.
Given that Jenna's boat is at a crest and Mel's boat is at its lowest point, we know that there are 4 additional crests between them. So, the total number of crests between Jenna and Mel is 4 + 1 = 5.
Since we know that the frequency of the waves is 31/min, we can determine the time it takes for one wave to pass a fixed point, which is the period (T), using the formula:
T = 1 / frequency
T = 1 / 31/min
We convert the frequency to seconds to match the unit of time:
T = 1 / (31/min) * (1 min / 60 sec)
Simplifying:
T = 1 / (31/60)
T = 60 / 31 sec
Now, we can calculate the distance between two consecutive crests using the formula:
wavelength (λ) = wave speed (v) * period (T)
The wave speed (v) can be calculated using the formula:
v = frequency (f) * wavelength (λ)
Since the wave speed is dependent on the properties of the medium (in this case, water), and the problem does not provide this information, we assume that the wave speed of the water waves is a constant and use an average value of 1.5 m/s.
Therefore, we have:
1.5 m/s = 31/min * λ
Now, we substitute the value of the period (T) and convert the frequency to seconds as before:
1.5 m/s = (31/60 sec) * λ
Simplifying:
1.5 m/s * 60/31 sec = λ
λ ≈ 2.90322580645 meters
Therefore, the wavelength of the water waves is approximately 2.903 meters.